The main claim of deriving special relativity from CA theory is in
section 9. They say:

: Special Relativity theory is founded on two basic postulates:: : (1) The velocity of light in a vacuum is constant and is equal for all: observers in inertial frames (inertial frame is one in which Newton?s: law of inertia is obeyed).:
: (2) The laws of physics are equally valid in all inertial reference
: frames.

which is OK. But they claim that there is no explanation of why (1) is
true in terms of (2). Actually from what I understand (1) follows

from
(2). The Maxwell equations for electromagnetism predict the existence
of EM waves which travel at a speed determined by various properties
of the vacuum. If you measure those properties the speed works out to
c, the speed of light. Hence Maxwell's equations predict that light
propagates at a speed of c. If you assume (2) then it will propagate
at c in all inertial frames, hence (1) follows.

Now, here is their explanation for (2) in terms of the universe as a CA:

: The second postulate of special relativity states that the laws of: physics are equally valid in all inertial reference frames. Stated: in a weaker form, there are no preferred reference frames to judge: absolute constant velocity motion (or inertial frames). This latter: form is easily explained in CA theory, by remembering that all cells: and their corresponding rules in the cellular automata are absolutely: identical everywhere. Motion itself is an illusion, and really represents: information transfers from cell to cell. To assign meaning to motion: in a CA, one must relate information pattern flows from one numeric: pattern group with respect to another group (the actual cell locations: are inaccessible to experiment). Therefore, motion requires reference: frames. Unless you have access to the absolute location of the cells,: all motion remains relative in CA theory. In other words, there is no
: reference frame accessible by experiment that can be considered as the
: absolute reference frame for constant velocity motion.

This is so vague as to be meaningless. There is no reference to any
properties of the CA other than that it can let patterns "flow" from
place to place (somewhat like gliders in Conway's Life). Does it really
follow from this very general property that the laws of physics would
be the same in all reference frames? Just because patterns can flow,
it follows that absolute velocity can't be detected, and that physics
is the same for all uniform observers? I don't think so.